Physics 214b-2008 Practice problems for exam 2

You should be able to do each of these problems (including all sub-parts) in about 20 minutes. (The first problem appears long, but is mostly multiple choice, so you should still be able to do the whole problem in 20 mintues.) If it takes you longer, you have not mastered the associated material thoroughly enough. Note: Of course, these problems don’t cover exactly the same topics as the questions on the real exam. Therefore, doing well on these practice problems does not insure that you are adequately prepared for the exam. However, doing poorly on these problems does mean that you need to study more.

Some integrals you may or may not need:

For the following, the parameter a is assumed >0:

2b. What is ? Is the system in a stationary state?

3. Show that any observable that is conserved and which has no explicit time dependence in the operator must commute with the Hamiltonian.

4. Show that .

5. Let us define the operator , and insert it into the inner product just before the like this: . (The sum in this inner product runs over all the energy eigenfunctions for the potential energy under consideration; these functions are assumed to form a complete set. The operator may look pretty strange to you, but it is very commonly used in quantum mechanics. Explain why the insertion of the operator does not change the value of . Hint: it is easiest to do this problem by thinking in terms of Hilbert space, and to work by analogy with dot products of ordinary vectors in three dimensions.